Regularity requirements for Sard's Theorem

Question

The most common formulation of Sard's Theorem is that for $f\in C^{n-m+1}(\mathbb R^n, \mathbb R^m)$ with $n\ge m$, the set $f(C_f)$ has Lebesgue measure 0, where $C_f=\{x, df(x)=0\}$.

Question. Is it possible to weaken the assumption (and keep the conclusion) by assuming only $f\in C^{n-m}(\mathbb R^n, \mathbb R^m)$, $f$ is $n-m+1$ times differentiable with a locally bounded $n-m+1$ derivative?

Answer 1

According to Bates, S. M. (1993). Toward a precise smoothness hypothesis in Sard’s theorem. Proceedings of the American Mathematical Society, 117(1), 279-283, the answer is yes:

• By Theorem 1 in that paper, it is sufficient that $(n-m)$-th derivative is Lipschitz.
• By Theorem 3, there are counterexamples with $\alpha$-Hölder $(n-m)$-th derivative for every $\alpha<1$.